Shake genus and slice genus

Lisa Piccirillo (UT Austin)

An important difference between high dimensional smooth manifolds and smooth four-manifolds is the ability to represent any middle dimensional homology class with a smoothly embedded sphere. For four-manifolds this is not always possible even among the simplest cases: four-manifolds $X_0(K)$, called $0$-traces, obtained by attaching an $0$-framed 2-handle to the 4-ball along a knot $K\in S^3$. The $0$ shake genus of $K$records the minimal genus of any smooth embedded generator of the second homology of $X_0(K)$ and is clearly bounded above by the slice genus of $K$. It is conjectured that the $0$-shake genus can be strictly less than the slice genus. We prove that slice genus is not a $0$-trace invariant, and thereby provide infinitely many examples of knots with $0$-shake genus strictly less than slice genus. This resolves problem 1.41 from the Kirby list. As a corollary we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but don't preserve slice genus.